In general this question \((\spadesuit )\)  has a negative answer. A counterexample can be found in [1]} as a “metric horn” which will be discussed later. See also [2]}, [3]}, [4]} for other examples along this direction. Let us emphasize that if (REF ) holds (as a non-collapsed sequence), then \(\mathfrak {m}=b \mathcal {H}^k\)  holds for some \(b \in (0, \infty )\) .
